Morgan–Mullen conjecture on primitive completely normal elements

Let $\mathbb F_{q^n}/\mathbb F_q$ be a finite field extension. The extension possesses the MM property if it contains an element $a\in\mathbb F_{q^n}$ that is primitive and completely normal. Morgan–Mullen conjecture. Every finite field extension possesses the MM property. This conjecture asks for primitive elements that are simultaneously normal over every intermediate field. It combines the Primitive Normal Basis Theorem with the Completely Normal Basis Theorem; the source presents it as a conjecture of Morgan and Mullen, with its resolution status not established here.